Why is Propensity Score called a distance measure as well?

I had to use Propensity score matching for my study. I used the MatchIt function in R and I studied what it actually does. I understand that the propensity score is calculated using Logistic regression and it is the probability (p) of the treatment ( or log[p/(1 − p)] ) which is used to balance the Case and control group, conditional upon the covariates. But I have read papers saying the propensity score is the same as a distance measure.. why is PS a distance? Aren't they slightly different? Shouldn't a distance be just an absolute difference of two propensity scores instead? (one from the Case and one from the Control group?) Also when I ran the MatchIt function and viewed the result after matching, the column (or variable) name for propensity scores was 'distance'... Am I understanding it wrong?

• I think it has to do with the fact that you compute the propensity of receiving a given treatment (say A instead of B), thus apparently you would compare two individuals according to their difference in propensity score, but eventually you need to factor in also their actual treatment, and thus propensity scores comprehensively takes into account the distance between predicted treatment and actual treatment, and thus is more a distance metric... – Joe_74 May 11 at 12:28
• ^ This is not why. "Distance" refers to the distance between two individuals, not between an individual's propensity score and the actual treatment received. Anything can be a distance score; it doesn't have to be related to the propensity score. – Noah May 13 at 5:10

The way MatchIt conceives of a distance score is the value for each individual that is used to compute the distance score as described previously. A precise way to describe this might be to call it a "position score," so that the difference between two position scores is a distance score. This is just an issue of terminology. Note that when you request Mahalanobis distance matching in MatchIt, no distance value is produced in the output. Clearly they are relying on this latter use of distance scores.
• The distance column is the propensity score (or whatever you chose to be the distance score, in the second sense). "Position score" is something I just made up right now, but yes, as I have used it, propensity scores would be position scores. That's not an actual term used. I was just pointing out that it might be a more intuitive way of thinking about it. – Noah May 13 at 5:09